Organisational aspects Topics

From syllogism to common sense:a tour through the logical landscape

Mehul Bhatt Oliver Kutz Thomas Schneider

Introduction 3 November 2011

Bhatt, Kutz, Schneider From syllogism to common sense: Intro 1

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And now . . .

1 Organisational aspects

2 Topics

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When, where, how?

Lecture Thursdays 16–18, MZH 1110Tutorial Thursdays 18–19, MZH 1110Room change: 3 Nov and 17 Nov in MZH 3150ECTS: 4Literature will be announced as we go alongLecture homepage: tinyurl.com/ws2011-logic

ExamAt the end of the semester, pick 2 topics from the course,present them to us in an oral exam and answer our questions.

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Who?

Mehul Bhatt bhatt

Oliver Kutz okutz

Thomas Schneider tschneider

@informatik.uni-bremen.de

Bhatt, Kutz, Schneider From syllogism to common sense: Intro 4

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And now . . .

1 Organisational aspects

2 Topics

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Aristotle’s Syllogisms

First formal study of logic in Aristotle’s Prior Analytics

Central elements: schemes such asIf all A are B If all men are mortaland all B are C e.g. and all Greeks are menthen all A are C then all Greeks are mortal

We willexplain and discuss valid types of syllogism,show their corresponding Venn diagrammes,explore common fallacies (reasoning mistakes with syllogisms),play with the Interactive Syllogistic Machine

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Propositional logic

The most basic classical logic: allows to combine atomicpropositions using Boolean operators (“and”, “or”, “not”)

Two notions of consequence/entailment:proof-theoretic (�) and semantic (|=)

� implies |=: soundness of the proof theory|= implies �: completeness

We’ll introduce basic terminology

We’ll also look at conditionals such asIf he takes a plane he will get here quicker.He will take a plane.Hence, he will get here quicker.

If New York is in New Zealand, then 2 + 4 = 4.

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Intuitionistic logicBrouwer’s Intuitionistic logic developed out of the crisis in thefoundations of mathematics in the late 19th centuryThe philosophy of intuitionism puts a stong emphasis on thenotion of (mental) construction. Its logic therefore

rejects the classical principle of tertium non datur (excludedmiddle) and double negation eliminationit is a restriction of classical logicnot truth is preserved, but justificationproof by contradiction can not be used, as it does not yield aconstruction

We will look atHeyting’s axiomatisation of intuitionistic propositional logicKripke frames as a semantics for intuitionistic logic in termsof ‘stages of knowledge’the Gödel translation establishing a correspondence betweenclassical and intuitionistic logic

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Modal logic

Extends propositional logicConcerned with the modes in which things can be true/false:

ϕ is necessary or possibleϕ is true sometimes in the future or always in the futurethe agent knows or believes ϕ

ϕ is provable

Different axiomatisations lead to different modal systemsPossible-world semanticsModal tableauxSoundness, completenessStrict conditional and non-normal modal logics

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Many-valued logic

Many-valued logics give up the idea that there are exactly 2truth-valuesThey evaluate formulae not only to true or false, but e.g. to:

true, false, or indeterminate (Łukasiewicz)true, false, neither true nor false, both true and false (Belnap)true, false, degrees of truth in the interval [0, . . . , 1] (Zadeh)etc.

Such logics need more abstract model theory than BooleanAlgebrasMany applicaton areas, such as database theory (SQL),uncertain reasoning, and paraconsistent reasoningWe discuss basics of model theory, truth tables,axiomatisations, and the relation to classical logic

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Description logic

Allows for describing terminological knowlegde, e.g.Elephant ≡ Mammal�∃hasBodypart.Trunk�∀hasColour.Greydefines elephants extensionally

Fraction � ∃hasLocation.Arm� ∃hasLocation.(Ulna � Radius � Humerus)

states background knowledge about bone fractions

DLs are restricted, well-behaved fragments of first-order logic

DL theories (ontologies) are used in biomedicine, healthcare,linguistics, physics, chemistry, engineering, Semantic Web, . . .

Sophisticated tools can automatically infer implicit knowledge(reasoning), explain inferences, extract modules for reusingontologies, integrate ontologies and databases, . . .

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First-order logic

Extend propositional logic:predicates, functions, Boolean operators, quantification

Used to formalise axiomatic systemsand other fundamental mathematical knowledge, e.g.

∀xyz(xRy ∧ yRz → xRz): relation R is transitive�∀n P(n)

�↔

�P(0) ∧ ∀n

�P(n) → P(n + 1)

��: induction

Zermelo-Fraenkel set theory, Peano arithmetic, . . .

Also used as an ontology language more expressive than DLsEnjoys important meta-logical properties, butis undecidable ❀ not as computationally well-behaved as DLsWe’ll cover syntax, semantics, deduction system(s),completeness, compactness, Löwenheim-Skolem theorem,first-order ontologies, tool demo

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